Bellman’s lost-in-a-forest problem for the golden gnomon
Let be the isosceles triangle whose equal sides have length , whose base angles are , and whose apex angle is ; this is the golden gnomon. For a convex forest , let an escape path be a rectifiable curve no congruent copy of which is contained in , and let denote the infimum of the lengths of escape paths for . Determine and an extremal path. Theorem 1.1 of the cited paper proves that, for the exact calibration parameters defined in Section 2,
The minimum is attained by the symmetric seven-piece path defined in equation (2), consisting of two segments, two circular arcs, two tangent segments, and one central truncating segment. Its exact length is
and the constant is transcendental.
References
Primary source
Progress summary
A July 2026 preprint claims the exact shortest escape route has been found, but the result has not yet been independently verified.
Bellman’s problem asks for the shortest route guaranteed to escape a golden-gnomon triangle from any starting position and direction. The preprint claims the optimum is a symmetric seven-piece path of straight segments, circular arcs, and tangents.
Known results
- Zalgaller’s construction gives the earlier benchmark, with scaled length .
- Gibbs’s unproved “Tunnel” phase predicted that a shorter seven-piece route should exist.
July 2026 claimed exact solution
The preprint states that the seven-piece route is optimal and has exact escape length , about below the Zalgaller benchmark. It characterizes using a uniquely isolated root of an explicit quartic and reports Lean verification of finite certificates. The authors say Claude Fable , GPT Sol, and Claude Opus assisted with the search, exposition, and formalization.
Current status (as of July 2026): An arXiv preprint claims a complete exact solution with optimum ; independent mathematical verification remains outstanding.
Solutions 0
No solutions have been posted yet.